I-automorphism: Difference between revisions

Revision as of 19:50, 9 August 2008

This article defines a property that can be evaluated for an automorphism of an algebra in a variety of algebras. The evaluation of that property depends on the ambient variety, and not just on the automorphism or the algebra.
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Definition

Suppose $V$ is a variety of algebras, and $A$ is an algebra in $V$ . An I-automorphism of $A$ is an automorphism that can be expressed as:

$x \mapsto φ (x, t_{1}, t_{2}, \dots, t_{n})$

where $t_{1}, t_{2}, \dots, t_{n} \in A$ are fixed, and $φ$ is a word in terms of the operations of the algebra,with the property that for any algebra of $V$ , and any choice of parameters, a map of the above form yields an automorphism.

Relation with other properties

Weaker properties

@@ Line 7: / Line 7: @@
 <math>x \mapsto \varphi(x,t_1,t_2,\dots,t_n)</math>
-where <math>\varphi</math> is a word in terms of the operations of the algebra, with the property that for ''any'' algebra of <math>\mathcal{V}</math>, a map of the above form yields an automorphism.
+where <math>t_1, t_2, \dots, t_n \in A</math> are fixed, and <math>\varphi</math> is a word in terms of the operations of the algebra,with the property that for ''any'' algebra of <math>\mathcal{V}</math>, and ''any'' choice of parameters, a map of the above form yields an automorphism.
 ==Relation with other properties==